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Chapter 2: Problem 18

Determine the following limits. $$\lim _{x \rightarrow-\infty} x^{-11}$$

Short Answer

Expert verified
Answer: The limit of the function \(f(x) = x^{-11}\) as x approaches negative infinity is 0.

Step by step solution

01

Define the function

Given the function \(f(x) = x^{-11}\), we want to find the limit as \(x\) approaches negative infinity. That is: $$\lim_{x \rightarrow -\infty} x^{-11}$$
02

Rewrite the function

The function can be rewritten as \(f(x) = \frac{1}{x^{11}}\). This will make it easier to conceptualize the behavior of the function for large negative values of \(x\): $$\lim_{x \rightarrow -\infty} \frac{1}{x^{11}}$$
03

Analyze the behavior

As \(x\) approaches negative infinity, we know that any positive power of \(x\), such as \(x^{11}\), approaches positive infinity. Therefore, the denominator of the fraction will grow towards positive infinity: $$x^{11} \rightarrow +\infty \quad \text{as} \quad x \rightarrow -\infty$$
04

Determine the limit

With the denominator, \(x^{11}\), approaching positive infinity, the value of the fraction, \(\frac{1}{x^{11}}\), tends towards zero. So, we have: $$\lim_{x \rightarrow -\infty} \frac{1}{x^{11}} = 0$$
05

Conclusion

The limit of the function \(f(x) = x^{-11}\) as \(x\) approaches negative infinity is 0: $$\lim_{x \rightarrow -\infty} x^{-11} = 0$$

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Most popular questions from this chapter

a. Evaluate \(\lim _{x \rightarrow \infty} f(x)\) and \(\lim _{x \rightarrow-\infty} f(x),\) and then identify any horizontal asymptotes. b. Find the vertical asymptotes. For each vertical asymptote \(x=a\), evaluate \(\lim _{x \rightarrow a^{-}} f(x)\) and \(\lim _{x \rightarrow a^{+}} f(x)\). $$f(x)=\frac{x-1}{x^{2 / 3}-1}$$

Show that the following functions have a removable discontinuity at the given point. See Exercises \(91-92\) $$f(x)=\frac{x^{2}-7 x+10}{x-2} ; x=2$$

If a function \(f\) represents a system that varies in time, the existence of \(\lim _{t \rightarrow \infty} f(t)\) means that the system reaches a steady state (or equilibrium). For the following systems, determine if a steady state exists and give the steady-state value. The amount of drug (in milligrams) in the blood after an IV tube is inserted is \(m(t)=200\left(1-2^{-t}\right)\).

Asymptotes Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions. $$h(x)=\frac{e^{x}}{(x+1)^{3}}$$

We write \(\lim _{x \rightarrow a} f(x)=-\infty\) if for any negative number \(M\) there exists \(a \delta>0\) such that $$f(x) < M \quad \text { whenever } \quad 0< |x-a| < \delta$$ Use this definition to prove the following statements. $$\lim _{x \rightarrow 1} \frac{-2}{(x-1)^{2}}=-\infty$$
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